Dispersionless limit of the Konno-Oono equation

Abstract

The study of integrable systems or solvable nonlinear differential equations (NDE) has been an active area of research since the discovery of the inverse scattering method. These equations are in a sense universal because they are found in many areas of physics and mathematics. By integrable systems, we mean those that have an infinite hierarchy of symmetries and conservation laws. There are several parallel construction schemes for the integrated systems. In addition to the integrable NDE, there is another important class of integrable partial differential equations: the socalled integrable hydrodynamic equations often called dispersionless equations. They often arise in various physics and mathematics problems and have been intensively studied in recent years. In this paper we investigate the coupled integrable dispersionless equation and its reduction. The dispersionless (quasiclassical) limit for the Konno-Oono equation is obtained and the Lax representation is constructed, which proves its integrability.